There are many current technological applications in which implementation of particular features may give rise to problems that may be expressed as optimization, estimation, and/or approximation problems. For example, implementing applications related to radar, communications, computer vision, optics, signal processing, distributed networks, etc., may give rise to technical implementation problems that may be expressed as unit-modulus least squares (UMLS) problems. UMLS problems can also be recast to a generalized form, namely, as unit-modulus quadratic program (UMQP) problems. In one particular example, which is discussed in more detail below, symbol detection in multiple-input multiple-output (MIMO) communications may involve estimating a signal vector, which may be expressed as a UMLS problem. Thus, implementing MIMO detection may give rise to technical implementation problems that may be expressed as a UMLS problem. Other particular examples of implementations that may involve signal processing problems that may be expressed as UMLS/UMQP problem expressions may include phase-only beamforming, source localization, phase synchronization, phase retrieval, etc.
Thus, solving these technical implementation problems expressed as UMLS/UMQP problem expressions may involve solving a generalized UMLS/UMQP problem. In general terms, a solution to a UMLS/UMQP problem may involve optimizing a multivariate least squares or quadratic function subject to the constraint that all variables have unit magnitudes. Some solutions have been proposed to solve the generalized UMLS/UMQP problems.
One such solution to the UMLS/UMQP problem is based on the semi-definite relaxation (SDR) approach. As will be discussed in further detail below, the general UMLS/UMQP problem is nonconvex and is also non-deterministic polynomial-hard (NP-hard) due to unit-modulus constraints. The SDR approach involves relaxing the nonconvex constraint to a convex optimization problem. However, SDR lifts the problem dimension and requires solving a much larger-scale convex problem. As such, an implementation using SDR will have a high computational complexity and may not be suitable for large-scale scenarios.
Another solution to the UMLS/UMQP problem that has been proposed is the gradient projection (GP) approach. Under the GP approach, as will be discussed in more detail below, at each iteration of the optimization, the solution is first updated along the negative gradient direction, and then the result is projected back to the constraint set to ensure all the iterates satisfy the unit-modulus requirement. However, the convergence rate of the GP approach may be slow. As such, any system implemented using GP may require a large number of iterations to obtain a satisfactory accuracy of results.